"No time, no space
another race of vibration,
the sea of the simulation"
(Franco Battiato - "No time, no space")
There are strange moments in life when, all of a sudden, one finds oneself writing a paper about loop quantum gravity. No, not even that. Mostly I am just drifting along the currents of the open sea, waiting for an improbable sighting of a distant shore or a passing ship. With no goal at hand, time passes in a mesmerizing dilation of slow movement and thoughts fail to coalesce around any solid object. Like the survivors portrayed in Théodore Géricault's "Le Radeau de la Méduse", I feel like I am ready to resort to cannibalizing my own work, just to get going with one apparent step in some random direction, while waiting for the shipwreck to run its course, back to firm land or to the ocean depths. So I am trying to get by these days, by reformatting in the language of loop quantum gravity some old thoughts. What for? Nothing, letting time pass.
In the loop quantum gravity approach, space-time is quantized by a procedure that encodes it in a discretized structure, consisting of spin networks and spin foams. A spin network consists of an oriented embedded graph in a 3-dimensional manifold with edges labelled by SU(2) representations and edges labelled by intertwiners between the representations attached to incoming and outgoing vertices. These representations relate to gravity in terms of holonomies of connections, and the formulation of Einstein's equations in terms of vierbein, or tetrads, and dual co-tetrads. Thus, to a spin networks, or the 1-skeleton of a triangulation by tetrahedra, one assigns operators of quantized area and volume, coming from counting intersection points of a surface, or 3-dimensional regions, with the edges or vertices of the spin network with a multiplicity given in terms of the spin representation attached to the edges and the intertwiners attached to the vertices.
This quantized version of embedded graphs and tetrahedra of a triangulation, developed within loop quantum gravity, gave rise to very interesting topological applications, such as the Turaev-Viro invariants of 3-manifolds. A spin foam is a 2-dimensional simplicial complex, which gives a geometric transition amplitude between two spin networks, and provides a "sum over histories" approach to loop quantum gravity. Like spin networks provide a formalism for quantized versions of 3-dimensional geometries, spin foams are the discretized version of 4-dimensional spacetimes. A "sum over geometries" weighted by the Einstein Hilbert action, as in a semiclassical Hartle-Hawking approach, becomes in this point of view a sum over spin foams, weighted by a "group field theory" type of action.
So far so good, and then what? I mean, what am I doing with all this other than trying to keep the raft afloat? I still don't know, mostly just playing around with it. I'll see where the flow goes, whether it is rip currents or peaceful stream. For those who seek a milder type of entertainment with the ideas of loop quantum gravity, there is always Greg Egan's novel "Schild's ladder", an action-adventure story of quantum gravity vacua and dynamical triangulations. Not a typical sci-fi bestseller, and sufficiently unusual to be genuinely entertaining.